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About this book :-
Elementary Calculus: An Infinitesimal Approach (2nd Edition) written by
H. Jerome Keisler .
This first-year calculus book is centered around the use of infinitesimals. It contains all the ordinary calculus topics, including the basic concepts of the derivative, continuity, and the integral, plus traditional limit concepts and approximation problems. Additional subjects include transcendental functions, series, vectors, partial derivatives, and multiple integrals. 2007 edition.
This book is concerned with the infinitesimal approach originally set forth by Newton and Leibnitz. The author has moved the theoretical material from Chapter One to an Appendix in this edition. A new chapter on differential equations has been added and the transcendental functions have been fully integrated into the first section. This book should be of interest to first and second year undergraduate mathematics students.
Book Detail :-
Title: Elementary Calculus: An Infinitesimal Approach
Edition: 2nd
Author(s): H. Jerome Keisler
Publisher: Brooks/Cole
Series: Dover Books on Mathematics
Year: 2012
Pages:
Type: PDF
Language: English
ISBN: 0486484521,9780486484525
Country: US
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About Author :-
The author Howard Jerome Keisler (born 3 December 1936) as an American mathematician, currently professor emeritus at University of Wisconsin–Madison. His research has included model theory and non-standard analysis.
Keisler published Elementary Calculus: An Infinitesimal Approach, a first-year calculus textbook conceptually centered on the use of infinitesimals, rather than the epsilon, delta approach, for developing the calculus.
He is also known for extending the Henkin construction (of Leon Henkin) to what are now called Henkin–Keisler models. In 2012 he became a fellow of the American Mathematical Society.
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Book Contents :-
Elementary Calculus: An Infinitesimal Approach (2nd Edition) written by
H. Jerome Keisler
cover the following topics.
INTRODUCTION
1. REAL AND HVPERREAL NUMBERS
1.1 The Real Line
1.2 Functions of Real Numbers
1.3 Straight Lines
1.4 Slope and Velocity; The Hyperreal Line
1.5 Infinitesimal, Finite, and Infinite Numbers
1.6 Standard Parts
Extra Problems for Chapter I
2. DIFFERENTIATION
2.1 Derivatives
2.2 Differentials and Tangent Lines
2.3 Derivatives of Rational Functions
2.4 Inverse Functions
2.5 Transcendental Functions
2.6 Chain Rule
2.7 Higher Derivatives
2.8 Implicit Functions
Extra Problems for Chapter 2
3. CONTINUOUS FUNCTIONS
3.1 How to Set Up a Problem
3.2 Related Rates
3.3 Limits
3.4 Continuity
3.5 Maxima and Minima
3.6 Maxima and Minima - Applications
3.7 Derivatives and Curve Sketching
3.8 Properties of Continuous Functions
Extra Problems for Chapter 3
4. INTEGRATION
4.1 The Definite Integral
4.2 Fundamental Theorem of Calculus
4.3 Indefinite Integrals
4.4 Integration by Change of Variables
4.5 Area between Two Curves
4.6 Numerical Integration
Extra Problems for Chapter 4
5. LIMITS, ANALYTIC GEOMETRY, AND APPROXIMATIONS
5.1 Infinite Limits
5.2 L
Hospital
s Rule
5.3 Limits and Curve Sketching
5.4 Parabolas
5.5 Ellipses and Hyperbolas
5.6 Second Degree Curves
5.7 Rotation of Axes
5.8 The e, 8 Condition for Limits
5.9 Newton
s Method
5.10 Derivatives and Increments
Extra Problems for Chapter 5
6. APPLICATIONS OF THE INTEGRAL
6.1 Infinite Sum Theorem
6.2 Volumes of Solids of Revolution
6.3 Length of a Curve
6.4 Area of a Surface of Revolution
6.5 Averages
6.6 Some Applications to Physics
6.7 Improper Integrals
Extra Problems for Chapter 6
7. TRIGONOMETRIC FUNCTIONS
7.1 Trigonometry
7.2 Derivatives of Trigonometric Functions
7.3 Inverse Trigonometric Functions
7.4 Integration by Parts
7.5 Integrals of Powers of Trigonometric Functions
7.6 Trigonometric Substitutions
7.7 Polar Coordinates
7.8 Slopes and Curve Sketching in Polar Coordinates
7.9 Area in Polar Coordinates
7.10 Length of a Curve in Polar Coordinates
Extra Problems for Chapter 7
8. EXPONENTIAL AND LOGARITHMIC FUNCTIONS
8.1 Exponential Functions
8.2 Logarithmic Functions
8.3 Derivatives of Exponential Functions and the Number e
8.4 Some Uses of Exponential Functions
8.5 Natural Logarithms
8.6 Some Differential Equations
8.7 Derivatives and Integrals Involving In x
8.8 Integration of Rational Functions
8.9 Methods of Integration
Extra Problems for Chapter 8
9. INFINITE SERIES
9.1 Sequences
9.2 Series
9.3 Properties of Infinite Series
9.4 Series with Positive Terms
9.5 Alternating Series
9.6 Absolute and Conditional Convergence
9.7 Power Series
9.8 Derivatives and Integrals of Power Series
9.9 Approximations by Power Series
9.10 Taylor
s Formula 547
9.11 Taylor Series 554
Extra Problems for Chapter 9
10. VECTORS
10.1 Vector Algebra
10.2 Vectors and Plane Geometry
10.3 Vectors and Lines in Space
10.4 Products of Vectors
10.5 Planes in Space
10.6 Vector Valued Functions
10.7 Vector Derivatives
10.8 Hyperreal Vectors
Extra Problems for Chapter I0
11. PARTIAL DIFFERENTIATION
II.1 Surfaces
11.2 Continuous Functions of Two or More Variables
11.3 Partial Derivatives
11.4 Total Differentials and Tangent Planes
11.5 Chain Rule
11.6 Implicit Functions
11.7 Maxima and Minima
11.8 Higher Partial Derivatives
Extra Problems for Chapter II
12. MULTIPLE INTEGRALS
12.1 Double Integrals
12.2 Iterated Integrals
12.3 Infinite Sum Theorem and Volume
12.4 Applications to Physics
12.5 Double Integrals in Polar Coordinates
12.6 Triple Integrals
12.7 Cylindrical and Spherical Coordinates
Extra Problems for Chapter 12
13. VECTOR CALCULUS
13.1 Directional Derivatives and Gradients
13.2 Line Integrals
13.3 Independence of Path
13.4 Green
s Theorem
13.5 Surface Area and Surface Integrals
13.6 Theorems of Stokes and Gauss
Extra Problems for Chapter 13
14. DIFFERENTIAL EQUATIONS
14.1 Equations with Separable Variables
14.2 First Order Homogeneous Linear Equations
14.3 First Order Linear Equations
14.4 Existence and Approximation of Solutions
14.5 Complex Numbers
14.6 Second Order Homogeneous Linear Equations
14.7 Second Order Linear Equations
Extra Problems for Chapter 14
APPENDIX: Tables
I Trigonometric Functions
II Greek Alphabet
III Exponential Functions
IV Natural Logarithms
V Powers And Roots
ANSWERS TO SELECTED PROBLEMS
Index
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